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How are public & private keys in an address created?

I am scanning in a private key from a QR code in an app I am writting to handle the import of physical bitcoins.

Given a private key, how can I get the public key so that I can lookup its value on the block explorer?

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In ECDSA public keys are usually generated from a random private key. The simple answer then would be, that instead of using a random private key, you use the given private key and it should generate the public key. After that, perform the steps listed in the Wiki to find out the conventional notation for the public key you can use anywhere.

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